Extension and Enhancement of the Allen-Eggers Analytic Solution for Ballistic Entry Trajectories

The closed-form analytic solution to the equations of motion for ballistic entry developed by Allen and Eggers is extended and enhanced with a method of choosing an appropriate constant flight-path angle, limits based on the equations of motion and acceptable ap-proximation error are proposed that bound the domain of applicability, and closed-form expressions for range and time-dependency. The expression developed for range to go exhibits error that may low enough for onboard drag-modulation guidance and targeting systems. These improvements address key weaknesses in the original approximate solution. Results show that the extended and enhanced Allen-Eggers solution provides good accu-racy across a range of ballistic coefficients entries at Earth with initial flight-path angles steeper than-7 deg. Nomenclature A = aerodynamic reference area, m2 a = acceleration magnitude, Earth g CD = drag coefficient CL = lift coefficient D = qACD, drag magnitude, N E = energy over weight, m g = acceleration due to gravity, m/s2 H = atmospheric scale height, m h = altitude, m k = convective stagnation-point heat rate constant, kg1/2m−1 L = qACL, lift magnitude, N L/D = lift-to-drag ratio m = mass, kg q = (1/2)ρV 2, dynamic pressure, N/m2 Q ̇ = convective stagnation-point heat rate, W/cm2 R = planetary radius, m rn = effective nose radius, m S = range, m Stogo = range to go, m t = time, s V = velocity magnitude, m/s VC = circular velocity, m/s W = mg, weight magnitude, N γ ̄ = Euler’s constant β = m/(CDA), ballistic coefficient, kg/m 2 δq = initial dynamic pressure stand-off factor δV = final velocity stand-off facto